Splitting: (1 + cos(x))/sin(x) = csc(x) + cot(x). ∫(csc(x) + cot(x)) dx = ln|tan(x/2)| + C.
(1 - cos(x))/sin(x) = tan(x/2). So ∫ tan(x/2) dx = -2 ln|cos(x/2)| + C = tan(x/2) + C (up to constant).
sin^3(x) = sin(x)(1 - cos^2(x)). Substitute u = cos(x), gives ∫ sin^3(x) dx = -cos(x) + cos^3(x)/3 + C.
cos^3(x) = cos(x)(1 - sin^2(x)), substitute u = sin(x). ∫ cos^3(x) dx = sin(x) - sin^3(x)/3 + C.
cos(x)/sin(x) = cot(x). ∫ cot(x) dx = ln|sin(x)| + C.
sin(x)/cos(x) = tan(x). ∫ tan(x) dx = -ln|cos(x)| + C.
1/sin^2(x) = csc^2(x). ∫ csc^2(x) dx = -cot(x) + C.
1/cos^2(x) = sec^2(x). ∫ sec^2(x) dx = tan(x) + C.
Split: (sin(x)/sin(x)cos(x)) + (cos(x)/sin(x)cos(x)) = 1/cos(x) + 1/sin(x). ∫ (sec(x) + csc(x)) dx = ln|tan(x)| + C.
cos^2(x)/sin(x) = (1 - sin^2(x))/sin(x) = csc(x) - sin(x). ∫ (csc(x) - sin(x)) dx = ln|tan(x/2)| + cos(x) + C.